{
  "abstract": "We give an approach to discover continued fractions for series of the form $$\\sum_{k=0}^\\infty \\frac{ε^k}{(x+k)^s},$$ where $ε= \\pm 1$. We find a continued fraction of the form \\begin{equation*} \\frac{a_1}{b_1(x)} \\fplus \\frac{a_2}{b_2(x)} \\fplus \\fdots \\end{equation*} where $a_k$ are constants and $b_k(x)$ are polynomials. Our technique involves telescoping continued fractions. This provides a discovery approach to continued fractions given by Ramanujan for $$2\\sum_{k=1}^\\infty \\frac{(-1)^{k+1}}{x+2k-1}, 2\\sum_{k=0}^\\infty \\frac{1}{(x+2k+1)^2}, 2\\sum_{k=0}^\\infty \\frac{(-1)^k}{(x+2k+1)^2}, \\sum_{k=1}^\\infty \\frac{1}{(x+k)^3}, $$ and the like. We display the first few terms of several continued fractions obtained in this manner for larger values of $s$, including $s=5, 7, 9, 11$. They do not follow as simple a pattern as Ramanujan's continued fractions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07092",
  "author_total": 2,
  "authors": [
    {
      "name": "Gaurav Bhatnagar"
    },
    {
      "name": "Krishnan Rajkumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/brmc4eskpzdk2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07092",
  "fulltext_url": null,
  "id": "brmc4eskpzdk2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07092",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212012710",
  "pdf_url": "https://arxiv.org/pdf/2609.07092",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/brmc4eskpzdk2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07092",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07092"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The method of telescoping continued fractions",
  "title_text": "The method of telescoping continued fractions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/brmc4eskpzdk2",
  "venue": "arXiv",
  "versions": []
}